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6.5 Shell Arithmetic
The shell allows arithmetic expressions to be evaluated, as one of
the shell expansions or by the let
and the ‘i’ option
to the declare
builtins.
Evaluation is done in fixedwidth integers with no check for overflow, though division by 0 is trapped and flagged as an error. The operators and their precedence, associativity, and values are the same as in the C language. The following list of operators is grouped into levels of equalprecedence operators. The levels are listed in order of decreasing precedence.

id++ id
variable postincrement and postdecrement

++id id
variable preincrement and predecrement

 +
unary minus and plus

! ~
logical and bitwise negation

**
exponentiation

* / %
multiplication, division, remainder

+ 
addition, subtraction

<< >>
left and right bitwise shifts

<= >= < >
comparison

== !=
equality and inequality

&
bitwise AND

^
bitwise exclusive OR


bitwise OR

&&
logical AND


logical OR

expr ? expr : expr
conditional operator

= *= /= %= += = <<= >>= &= ^= =
assignment

expr1 , expr2
comma
Shell variables are allowed as operands; parameter expansion is performed before the expression is evaluated. Within an expression, shell variables may also be referenced by name without using the parameter expansion syntax. A shell variable that is null or unset evaluates to 0 when referenced by name without using the parameter expansion syntax. The value of a variable is evaluated as an arithmetic expression when it is referenced, or when a variable which has been given the integer attribute using ‘declare i’ is assigned a value. A null value evaluates to 0. A shell variable need not have its integer attribute turned on to be used in an expression.
Constants with a leading 0 are interpreted as octal numbers.
A leading ‘0x’ or ‘0X’ denotes hexadecimal. Otherwise,
numbers take the form [base#
]n, where the optional base
is a decimal number between 2 and 64 representing the arithmetic
base, and n is a number in that base. If base#
is
omitted, then base 10 is used.
The digits greater than 9 are represented by the lowercase letters,
the uppercase letters, ‘@’, and ‘_’, in that order.
If base is less than or equal to 36, lowercase and uppercase
letters may be used interchangeably to represent numbers between 10
and 35.
Operators are evaluated in order of precedence. Subexpressions in parentheses are evaluated first and may override the precedence rules above.
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